Theorem 4.3.4Let Ω be a connected open set and let f : Ω → ℝpbe continuous. Then f has a potential Fif and only if

∫
f ⋅dγ
γ

is path independent for allγa bounded variation curve such thatγ∗is contained in Ω. This means theabove line integral depends only onγ

(a)

andγ

(b)

.

Proof: The first part was proved in Theorem 4.3.1. It remains to verify the existence of a potential in
the situation of path independence.

Let x0∈ Ω be fixed. Let S be the points x of Ω which have the property there is a bounded variation
curve joining x0 to x. Let γx0x denote such a curve. Note first that S is nonempty. To see this,
B

(x0,r)

⊆ Ω for r small enough. Every x ∈ B

(x0,r)

is in S. Then S is open because if x ∈ S, then
B

(x,r)

⊆ Ω for small enough r and if y ∈ B

(x,r)

, you could go take γx0x and from x follow the straight
line segment joining x to y. In addition to this, Ω ∖S must also be open because if x ∈ Ω ∖S, then choosing
B

(x,r)

⊆ Ω, no point of B

(x,r)

can be in S because then you could take the straight line segment
from that point to x and conclude that x ∈ S after all. Therefore, since Ω is connected, it
follows Ω ∖ S = ∅. Thus for every x ∈ S, there exists γx0x, a bounded variation curve from x0 to
x.

Define

∫
F (x) ≡ f ⋅dγx0x
γx0x

F is well defined by assumption. Now let lx

(x+tek)

denote the linear segment from x to x + tek. Thus
to get to x + tek you could first follow γx0x to x and from there follow lx

(x+tek)

to x + tek.
Hence

∫
F-(x+tek)−-F-(x)= 1 f ⋅dl
t t lx(x+te ) x(x+tek)
k

1∫ t
= t 0 f (x + sek) ⋅ekds → fk (x)

by continuity of f. Thus ∇F = f. ■

Corollary 4.3.5Let Ω be a connected open set and f : Ω → ℝp. Then f has a potential if andonly if every closed,γ

(a)

= γ

(b)

, bounded variation curve contained in Ω has the propertythat

∫
f ⋅dγ = 0
γ

Proof: Using Lemma 4.2.9, this condition about closed curves is equivalent to the condition
that the line integrals of the above theorem are path independent. This proves the corollary.
■

Such a vector valued function is called conservative. Summarizing the above we have the following
major theorem which is called the fundamental theorem of line integrals.